Extrapolation technique from mixed and variational estimators

The one body density matrix (3.84) corresponds to a mixed estimator, when the averaging of the variable $A$ is done in an asymmetric way $\langle\phi_0\vert\hat A\vert\psi_T\rangle$. If the trial wavefunction is close to the true ground-state wavefunction $\phi_0$ one can estimate the ground-state average $\langle\phi_0\vert\hat A\vert\psi_0\rangle$ by using the following technique. Let us denote the difference between the trial wave function and ground-state wave function as $\delta \psi$

$$\phi_0 = \psi_T + \delta \psi$$ (3.89)

Then the ground-state average can be written as

$$\langle\phi_0\vert\hat A\vert\phi_0\rangle = \langle\psi_T\vert\h... ...at A\vert\delta\psi\rangle + \langle\delta\psi\vert\hat A\vert\delta\psi\rangle$$ (3.90)

If $\delta \psi$ is small the second order term $\langle\delta\psi\vert\hat A\vert\delta\psi\rangle$ can be neglected. After substitution $\langle\phi_0\vert\hat A\vert\delta\psi\rangle = \langle\psi_T\vert\hat A\vert\phi_0\rangle - \langle\psi_T\vert\hat A\vert\psi_T\rangle$ the extrapolation formula becomes

$$\langle A\rangle = \langle\phi_0\vert\hat A\vert\phi_0\rangle = 2... ...phi_0\vert\hat A\vert\psi_T\rangle - \langle\psi_T\vert\hat A\vert\psi_T\rangle$$ (3.91)